Multivariate Splines and the 4 Color Map Problem

Notes

This slide lists all the configurations making up an unavoidable
set simply due to the observation that there must be a vertex of
degree less than 6.

The vertex whose star we consider is marked green. Red
edges may be shared with the rest of the triangulation.
Thus the first three configurations are the stars of
interior vertices of degree 3, 4, or 5. The remaining
configurations may arise as the stars of boundary vertices
of degree 2, 3, 4, or 5.

However, in the proof of the dimension statement for S14 we
consider subtriangulation, and so the first three
configurations in the second row may also arise as
subtriangulations of the star of an interior vertex, part of
which has already been removed.